# AdaptiveIntegrator Class

## Definition

**Namespace:**Extreme.Mathematics.Calculus

**Assembly:**Extreme.Numerics (in Extreme.Numerics.dll) Version: 8.1.23

`public class AdaptiveIntegrator : NumericalIntegrator`

- Inheritance
- Object → ManagedIterativeAlgorithm<Double, Double, SolutionReport<Double, Double>> → ManagedIterativeAlgorithm<Double> → IterativeAlgorithm → NumericalIntegrator → AdaptiveIntegrator

## Remarks

The AdaptiveIntegrator class provides the most robust, reliable, and in most cases the fastest numerical integration algorithm. It uses a simple integration rule to approximate the integral over subintervals. In each step, the interval with the largest estimated error is divided in two, and new approximations for these two subintervals are calculated. This process continues until the total estimated error over all intervals is within the desired tolerance.

AdaptiveIntegrator
inherits from NumericalIntegrator, the
abstract base class for all numerical integration classes.
It, in turn, inherits from the IterativeAlgorithm
class. All properties of this class are also available.
The AbsoluteTolerance and
RelativeTolerance properties set the desired
precision as specified by the
ConvergenceCriterion property. The default
value for both tolerances is
SqrtEpsilon (roughly
10^{-8}). MaxIterations sets the
maximum number of iterations, which in this case is the
maximum number of subintervals. The default value for
this property is 5000. IterationsNeeded
returns the actual number of iterations performed after
the algorithm has completed.

The Integrate(Func<Double, Double>, Double, Double, Double[]) method does the actual work of numerically integrating an integrand. It takes three parameters. The first parameter is a function of one variable that specifies the function to integrate. The second and third parameters are Double values that specify the lower and upper bounds of the integration interval.

The IntegrationRule property lets you specify which integration rule to use to approximate integrals on a subinterval. Integration rules can be obtained by calling static methods of the IntegrationRule class. The default is the 31-point rule for normal integrands, and the 15-point rule when singularities are expected, and when integrating over infinite intervals. For oscillating integrands, the higher order rules will tend to give better results.

Several enhancements make this the integration algorithm of choice, especially for 'difficult' target functions.

An extra property, UseExtrapolation,
lets you specify whether special precautions should be
taken to deal with singularities. If UseExtrapolation
is **true**, an extrapolation algorithm is used to
approximate the integral near a singularity or
singularities. Because this option requires extra
resources, it is recommended only for integrands
that actually have singularities inside or at the
boundaries of the integration interval.

The SetSingularities(Double[]) and GetSingularities() methods let you specify and retrieve the points inside the integration interval where the integrand has a singularity or a discontinuity. They are returned as a Double array containing the problem points. Extrapolation is used once again to approximate the integral near the singularity or singularities. Specifying the problem points in the target function can lead to speed-ups of several orders of magnitude.

Unlike other classes, this integrator can deal successfully with infinite integration intervals. Specify NegativeInfinity or PositiveInfinity for one or both bounds of the integration interval. Internally, the class transforms the infinite interval to a finite interval, and performs the integration of the transformed integrand over the transformed interval.

## Constructors

Adaptive | Constructs a new AdaptiveIntegrator object. |

Adaptive | Constructs a new AdaptiveIntegrator object. |

Adaptive | Constructs a new AdaptiveIntegrator object. |

Adaptive | Constructs a new AdaptiveIntegrator object. |

Adaptive | Constructs a new AdaptiveIntegrator object. |

## Properties

Absolute |
Gets or sets the absolute tolerance used in the
convergence test.
(Inherited from IterativeAlgorithm) |

Convergence |
Gets or sets a value specifying the criterion that is
to be used in the convergence test for the algorithm.
(Inherited from IterativeAlgorithm) |

Convergence |
Gets the convergence test for the algorithm.
(Inherited from IterativeAlgorithm) |

Convergence |
Gets the collection of convergence tests for the algorithm.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Current |
Gets the estimated error of the approximation of the result
at the current iteration.
(Inherited from IterativeAlgorithm) |

Current |
Gets the approximation of the result at the current iteration.
(Inherited from IterativeAlgorithm) |

Estimated |
Gets a value indicating the size of the absolute
error of the result.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Evaluations |
Gets the number of evaluations needed to execute the algorithm.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Evaluations |
Gets the number of evaluations still available.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Has |
Indicates whether the degree of parallelism is a property that is shared
across instances.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Integrand |
Gets or sets the function to integrate.
(Inherited from NumericalIntegrator) |

Integration | Gets or sets the integration rule to be used to estimate integrals over subintervals. |

Iterations |
Gets the number of iterations needed by the
algorithm to reach the desired accuracy.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Iterations |
Gets the number of iterations remaining.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Lower |
Gets or sets the lower bound of the integration interval.
(Inherited from NumericalIntegrator) |

Max |
Gets or sets the maximum degree of parallelism enabled by this instance.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Max |
Gets or sets the maximum number of evaluations during the calculation.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Max | Gets or sets the maximum number of iterations
to use when approximating the roots of the target
function.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Min |
Gets or sets the minimum iterations that have to be performed.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Order |
Gets the order of the Gauss-Kronrod formula used to
calculate the integral on subintervals.
(Overrides NumericalIntegrator.Order) |

Parallel |
Gets or sets the configuration for the parallel behavior of the algorithm.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Relative |
Gets or sets the relative tolerance used in the
convergence test.
(Inherited from IterativeAlgorithm) |

Result |
Gets the result of an algorithm after it has executed.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Solution |
Gets the result of an algorithm after it has executed.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Status |
Gets the AlgorithmStatus following
an execution of the algorithm.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Throw |
Gets or sets a value indicating whether to throw an
exception when the algorithm fails to converge.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Transformed |
Gets or sets the
(Inherited from NumericalIntegrator) |

Upper |
Gets or sets the upper bound of the integration interval.
(Inherited from NumericalIntegrator) |

Use | Gets or sets a value indicating whether precautions should be taken to deal with singularities on the integration interval. |

## Methods

Clone |
Returns a copy of this numerical integrator object.
(Overrides NumericalIntegrator.Clone()) |

Equals | Determines whether the specified object is equal to the current object. (Inherited from Object) |

Error |
Returns the estimated error for testing whether the algorithm has converged.
(Inherited from IterativeAlgorithm) |

Evaluate |
Evaluates the integrand and increments the number of function evaluations.
(Inherited from NumericalIntegrator) |

Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object) |

Get | Serves as the default hash function. (Inherited from Object) |

Get | Returns an array of Double values containing the locations of singularities of the integrand inside the integration interval. |

Get | Gets the Type of the current instance. (Inherited from Object) |

Increment |
Increments the number of evaluations by one.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Increment |
Increments the number of evaluations by the specified amount.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Integrate() |
Numerically integrates a function of one variable.
(Inherited from NumericalIntegrator) |

Integrate( |
Numerically integrates a function of one variable.
(Inherited from NumericalIntegrator) |

Integrate( |
Numerically integrates a function of one variable.
(Inherited from NumericalIntegrator) |

Integrate( |
Numerically integrates a function of one variable.
(Inherited from NumericalIntegrator) |

Integrate( |
Numerically integrates a function of one variable.
(Inherited from NumericalIntegrator) |

Integrate( | Numerically integrates a function of one variable. |

Integrate( |
Numerically integrates a function of one variable.
(Inherited from NumericalIntegrator) |

Iterate |
Performs one iteration of the algorithm.
(Overrides ManagedIterativeAlgorithm<T, TError, TReport>.Iterate()) |

Iterated |
Performs tasks after the iteration is completed, but before
the status of the algorithm is finalized.
(Overrides IterativeAlgorithm.Iterated()) |

Memberwise | Creates a shallow copy of the current Object. (Inherited from Object) |

OnConvergence |
Performs any tasks after the main algorithm has converged.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

OnFailure |
Performs any tasks after the main algorithm has failed to converge.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

OnInit |
Initializes the algorithm.
(Overrides NumericalIntegrator.OnInit()) |

Report |
Reports that the algoritm has failed. No further iterations will be performed.
(Inherited from IterativeAlgorithm) |

Report |
Records the results of an algorithm in case it fails.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Report |
Records the results of an algorithm.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Report |
Reports that the algoritm has succeeded. No further iterations will be performed.
(Inherited from IterativeAlgorithm) |

Report |
Records the results of a algorithm that converged successfully.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Reset |
Resets the number of evaluations to zero.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Restart |
Prepares the algorithm to be run again with possibly different inputs.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Run() |
Runs the algorithm.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Run( |
Runs the algorithm using the specified parallelization options.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Set |
Sets the results of an algorithm's execution.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Set | Sets the locations of singularities of the integrand inside the integration interval. |

Test |
Checks whether the algorithm has converged.
(Inherited from NumericalIntegrator) |

Thread |
Increments the number of evaluations by one.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Thread |
Increments the number of evaluations by the specified amount.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

Throw |
Interprets the AlgorithmStatus and
throws the appropriate exception.
(Inherited from ManagedIterativeAlgorithm<T, TError, TReport>) |

ToString | Returns a string that represents the current object. (Inherited from Object) |

Update |
Updates the current approximation in a point-wise convergence.
(Inherited from IterativeAlgorithm) |

Value |
Returns the value to compare the estimated error to
when testing whether the algorithm has converged.
(Inherited from IterativeAlgorithm) |